Table 1 SEIR models generated by various distributions with delay effect in the previous papers

Dirac-delta \(P_{D}(t)=\big\{\begin{array}{ll} 1, &u\in [0,\tau ),\\ 0 ,& u \in [\tau ,\infty ), \end{array}\) for τ>0, constant

\(\frac{\mathrm {d}E(t)}{\mathrm {d}t}=\beta (t) S(t)I(t) - \beta (t-\tau ) S(t-\tau )I(t-\tau )\exp (-\mu \tau ) -\mu E(t)\), \(\frac{\mathrm {d}I(t)}{\mathrm {d}t} = \beta (t-\tau )S(t-\tau )I(t-\tau )\exp (-\mu \tau ) -(\gamma +\mu ) I(t)\)

\(\frac{\bar{\beta }\exp (-\mu \tau )}{\gamma +\mu }\)

System of DDEs, exact time of “delay”, NOT distributed

Exponential \(P_{E}(t)=\exp (-\lambda t)\)

\(\frac{\mathrm {d}E(t)}{\mathrm {d}t}=\beta (t) S(t)I(t) - (\lambda +\mu )E(t)\), \(\frac{\mathrm {d}I(t)}{\mathrm {d}t}=\lambda E(t)-(\gamma +\mu )I(t)\)

\(\frac{\bar{\beta }\mu }{(\lambda +\mu )(\gamma +\mu )}\)

System of simple ODEs, memoryless property

Gamma \({P_{G}(u)=\sum_{i=0}^{n-1} \frac{1}{i!} e^{-n\lambda u}(n\lambda u)^{i}}\), for n, positive integer

\(\frac{\mathrm {d}E_{n}(t)}{\mathrm {d}t}= \beta (t) S(t)I(t)-(n\lambda +\mu )E_{n} \), \(\frac{\mathrm {d}E_{i}(t)}{\mathrm {d}t}= n\lambda E_{i+1}-(n\lambda +\mu )E_{i}\), for i = n − 1,…,2,1, \(\frac{\mathrm {d}I(t)}{\mathrm {d}t}=n\lambda E_{1}(t) -(\gamma +\mu ) I(t)\), with \(E(t)=\sum_{i=0}^{n-1} E_{n-i}(t)\)

\(\frac{\bar{\beta }}{\gamma +\mu } (\frac{n\lambda }{n\lambda +\mu } )^{n}\)

Linear chains, unimodal, short-tailed distribution, can use “average” concepts!

Mittag-Leffler \(P_{M}(t)=E_{\alpha ,1} (-(t/\zeta )^{\alpha } )\), for 0<α ≤ 1 where \(E_{\alpha ,\beta }(z)=\sum_{k=0}^{\infty }\frac{z^{k}}{\varGamma (\alpha k+\beta )}\)

\(\frac{\mathrm {d}E(t)}{\mathrm {d}t}=\beta (t) S(t)I(t) - \exp (-\mu t)\zeta ^{-\alpha }{}_{0}\mathcal{D}_{t}^{1-\alpha } [\exp (\mu t)E(t) ] -\mu E(t)\), \(\frac{\mathrm {d}I(t)}{\mathrm {d}t}=\exp (-\mu t)\zeta ^{-\alpha }{}_{0}\mathcal{D}_{t}^{1-\alpha } [\exp (\mu t)E(t) ] -(\gamma +\mu ) I(t)\), where \({}_{0}\mathcal{D}_{t}^{1-\alpha }[f(t)]=\frac{1}{\varGamma (\alpha )} \frac{\mathrm {d}}{\mathrm {d}t} \int _{0}^{t} (t-u)^{\alpha -1}f(u) \,\mathrm {d}u\)

\(\frac{\bar{\beta }}{\gamma +\mu } \cdot \frac{1}{1+(\zeta \mu )^{\alpha }}\)

System of FDEs, heavy-tailed distribution, hard to get exact form of distribution, hard to find physical meanings of α